Understand Finance - Finance Tutorials and Finance How To's

Do you have a car payment to make each month? Do you pay rent on
an apartment? Both of these are annuities! As you may have
guessed, an annuity is simply a series of equal cash flows equally
spaced out in time. There are two main types of annuities–ordinary
annuities and annuities due.

An ordinary annuity is one in which the cash flows occur at the
end of the time period. For example, if you start a new job with a
monthly salary of $3,000, you receive your check for this amount at
the end of the month. The fact that the cash flow (your salary in
this case) is received at the end of the period along with the fact
that it is the same value each month makes this an ordinary
annuity.

So what in the world is an annuity due? It is just a
series of equal cash flows that occur at the beginning of
the period. My guess is, you’re probably sufferring through an
annuity due as we speak. If you are living in an apartment, making
those rent payments, you’re paying an annuity due. When you pay
your rent on the first of the month, you’re paying for the month
ahead of time. That is, on August 1st, you pay rent for the right
to live in the apartment for the rest of the month of August. This
is an annuity due because the payments occur at the
beginning of the period.

Let’s look at a simple example of an ordinary annuity. If you
deposit $1,000 at the end of each year into an account earning 9%
compounded annually, how much will you have in the end of 5 years?
Lets look at this on a timeline:

On our timeline, we see the $1,000 payments going into our
account. The numbers of the bottom refer to the end of the year. I
think after drawing the timeline, we’re ready to put this into our
financial calculator. We have 5 years, 9% interest, no present
value (see the timeline at time period 0?), and $1,000
payments.

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 5 Step 4. Set I/Y = 9% Step 5. Set PMT = -1,000 Step 6. Compute FV

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Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 5 Step 4. Set I/YR = 9% Step 5. Set PMT = -1,000 Step 6. Compute FV

If all went well, your calculator should have given you a future
value at time period 5 of $5,984.71. You may be
wondering why we put the $1,000 payments in as a negative number.
The reason for this is that these are cash outflows. A
good rule of thumb is that if money is leaving your wallet, we put
it in our calculator as a negative number. When money is coming
back to you, it is a positive number. This problem also taught us a
great lesson in that when we deal with ordinary annuities, the
calculator will give us our future values in the final year of the
annuity. In other words, the future value we just calculated is at
year 5 on our timeline.

Now let’s look at an annuity due (begin mode) problem. How much
would you be willing to pay for an annuity that paid you $1,000 at
the beginning of the year for 3 years if your opportunity cost is
11%?

Normally, the word “begin” makes your heart race and your hands
sweaty. It really isn’t that bad though. Again, we draw a
timeline.

The first thing that you should notice about the timeline above
is that we start at year 1 instead of year 0. When we’re dealing
with an annuity due, the numbers on our timeline represent the
beginning of the year. So the 1 tells you that the $1,000 payment
occurs at the beginning of year 1. We’re going to handle
this annuity due the same way we would handle an ordinary annuity.
The only difference is that we will set our calculator to begin
mode.

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Texas Instruments BAII Plus

Step 1. Begin Mode Step 2. Clear the calculator: Step 3. Annual compounding: Step 4. Set N = 3 Step 5. Set I/Y = 11% Step 6. Set PMT = 1,000 Step 7. Compute PV

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Hewlett-Packard 10BII

Step 1. Begin Mode Step 2. Clear the calculator: Step 3. Annual compounding: Step 4. Set N = 3 Step 5. Set I/YR = 11% Step 6. Set PMT = 1,000 Step 7. Compute PV

Piece of cake! Your calculator shows you that you’d be willing
to pay $2,712.52 today for this annuity due.
You’ve gotten the hang of annuities now, so try these practice
problems on your own.

Practice Problem 1
If you deposit $100 at the end of each month into an account
earning 11.5% annually, how much will you have saved at the end of
25 years?

Solution
Wow! You’ve saved up $172,011.55. If you’re not
getting the same answer that I do, make sure your calculator isn’t
in begin mode.

Practice Problem 2
How much would you have
to deposit at the beginning of each month to have $10,000 saved at
the end of the year? Assume your account earns 4% compounded
annually.

Solution
Only $815.45 a
month!